In the field of integrated circuit (“IC”) design and particularly very large scale integration (“VLSI”) design, it is desirable to test the design before implementation and to identify potential violations in the design. Before implementation on a chip, the information about a design, including information about specific signals and devices that comprise the design, as well as information about connections between the devices, are typically stored in a computer memory. Based on the connection and device information, the designer can perform tests on the design to identify potential problems. For example, one portion of the design that might be tested is the conducting material on the chip. In particular, representations of individual metal segments may be analyzed to determine whether they meet certain specifications, such as electromigration and self-heating specifications. Other tests that may be conducted include electrical rules checking tests, such as tests for noise immunity and maximum driven capacitance, and power analysis tests that estimate power driven by a particular signal and identify those over a given current draw. These tests may be performed using software tools referred to as VLSI circuit analysis tools.
Modern semiconductor IC chips include a dense array of narrow, thin-film metallic conductors, referred to as “interconnects”, that transport current between various devices on the IC chip. As the complexity of ICs continues to increase, the individual components must become increasingly reliable if the reliability of the overall IC is to be maintained. Due to continuing miniaturization of VLSI circuits, thin-film metallic conductors are subject to increasingly high current densities. Under such conditions, electromigration can lead to the electrical failure of interconnects in a relatively short period of time, thus reducing the lifetime of the IC to an unacceptable level. It is therefore of great technological importance to understand and control electromigration failure in thin film interconnects.
Electromigration can be defined as migration of atoms in a metal interconnect line due to momentum transfer from conduction electrons. The metal atoms migrate in the direction of current flow and can lead to failure of the metal line. Electromigration is dependent on the type of metal used and correlates to the melting temperature of the metal. In general, a higher melting temperature corresponds to higher electromigration resistance. Electromigration can occur due to diffusion in the bulk of the material, at the grain boundaries, or on the surface. For example, electromigration in aluminum occurs primarily at the grain boundary due to the higher grain boundary diffusivity over the bulk diffusivity and the excellent surface passivation effect of aluminum oxide that forms on the surface of aluminum when it is exposed to oxygen. In contrast, copper exhibits little electromigration in the bulk and at the grain boundary and instead primarily exhibits electromigration on the surface due to poor copper oxide passivation properties.
Electromigration can cause various types of failures in narrow interconnects, including void failures along the length of a line and diffusive displacements at the terminals of a line that destroy electrical contact. Both types of failure are affected by the microstructure of the line and can therefore be delayed or overcome by metallurgical changes that alter the microstructure. As previously noted, electromigration is the result of the transfer of momentum from electrons moving in an applied electric field to the ions comprising the lattice of the interconnect material. Specifically, when electrons are conducted through a metal, they interact with imperfections in the lattice and scatter. Thermal energy produces scattering by causing atoms to vibrate; the higher the temperature, the more out of place the atom is, the greater the scattering, and the greater the resistivity. Electromigration does not occur in semiconductors, but may in some semiconductor materials that are so heavily doped as to exhibit metallic conduction.
The driving forces behind electromigration are “direct force”, which is defined as the direct action of the external field on the charge of the migrating ion, and “wind force”, which is defined as the scattering of the conduction electrons by the metal atom under consideration. For simplicity, “electron wind force” often refers to the net effect of these two electrical forces. This simplification will also be used throughout the following discussion. These forces and the relation therebetween are illustrated in FIG. 1.
The electromigration failure process is predominantly influenced by the metallurgical-statistical properties of the interconnect, the thermal accelerating process, and the healing effects. The metallurgical-statistical properties of a conductor film refer to the microstructure parameters of the conductor material, including grain size distribution, the distribution of grain boundary misorientation angles, and the inclinations of grain boundaries with respect to electron flow. The variation of these microstructural parameters over a film causes a non-uniform distribution of atomic flow rate. Non-zero atomic flux divergence exists at the places where the number of atoms flowing into the area is not equal to the number of atoms flowing out of that area per unit time such that there exists either a mass depletion (divergence>0) or accumulation (divergence<0), leading to formation of voids and hillocks, respectively. In such situations, failure results either from voids growing over the entire line width, causing line breakage, or from extrusions that cause short circuits to neighboring lines.
The thermal accelerating process is the acceleration process of electromigration damage due to a local increase in temperature. A uniform temperature distribution along an interconnect is possible only absent electromigration damage. Once a void is initiated, it causes the current density to increase in the area around the void due to the reduction in the cross-sectional area of the conductor. The increase of the local current density is referred as “current crowding.” Since joule heating, or “self-heating”, is proportional to the square of current density, the current crowding effect leads to a local temperature rise around the void that in turn further accelerates the void growth. The whole process continues until the void is large enough to result in a line break.
Healing effects are the result of atomic flow in the direction opposite to the electron wind force, i.e., the “back-flow,” during or after electromigration. The back-flow of mass is initiated once a redistribution of mass has begun to form. Healing effects tend to reduce the failure rate during electromigration and partially heals the damage after current is removed. Nonhomogenities, such as temperature and/or concentration gradients, resulting from electromigration damage are the cause of the back-flow.
The effects of electromigration may be slow to develop; however, if an electromigration problem exists, the progress toward a fault is inexorable. The results of an electromigration problem are illustrated in FIGS. 2 and 3. Before current is applied to a section of an IC chip that is first powered up, the metal comprising the interconnects thereof is uniformly distributed, as illustrated in FIG. 2, which illustrates a side view of an interconnect 200. However, in a section of metal that is at risk for electromigration, the mass transport of metal, which occurs in the direction of average current, represented in FIG. 3 by an arrow 301, results in metal moving from a first end 302a of the section to a second end 302b thereof. At some future time, depending on the amount of current flowing through and the thickness of the interconnect 200, electromigration will result in the formation of a void 304 at the first end 302a and a hillock 304 at the second end 302b. Eventually, as previously described, this migration of metal from one end of the wire to the other will result in a failure of the interconnect 200.
As also previously noted, self-heating contributes to the electromigration and actually affects the surrounding wires as well. As a wire carries current, it will heat up, thereby lowering the limits for electromigration in surrounding wires as well as the wire under consideration. It is important, therefore, to consider the effects of both electromigration and self-heating (collectively “EM/SH”) when analyzing and verifying the reliability of an IC chip design.
Typically, circuit designs are provided in a hierarchical organization that allows designers to leverage common blocks in multiple areas of the design. On the other hand, conventional circuit analysis tools (including, e.g., the EM/SH analysis tools) operate on all instantiations of circuits in a hierarchical design independently, even where multiple instantiations of a particular block are encountered. It would therefore be advantageous to increase runtime efficiency of an analysis tool by addressing this drawback.